For a wide class of Collatz-type maps, the paper characterizes nonzero periodic points as integer values of a (p,q)-adic interpolation function and recasts the periodic-point question as a translate-density question, under hypotheses the Collatz map satisfies.
Non-Archimedean Duality: Algebras, Groups, and Multipliers
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abstract
We develop a duality theory for multiplier Banach-Hopf algebras over a non-Archimedean field K. As examples, we consider algebras corresponding to discrete groups and zero-dimensional locally compact groups with K-valued Haar measure, as well as algebras of operators generated by regular representations of discrete groups.
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2024 1verdicts
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$\left(p,q\right)$-adic Analysis and the Collatz Conjecture
For a wide class of Collatz-type maps, the paper characterizes nonzero periodic points as integer values of a (p,q)-adic interpolation function and recasts the periodic-point question as a translate-density question, under hypotheses the Collatz map satisfies.